Take Problem 12. It asks you to prove that three distinct points lie on a single line. You look at the diagram. They don't look like they line up. They look chaotic. The skeptic in you says it is impossible; the geometer in you knows that intuition is a liar. This is the first lesson of the 106: the eyes are not to be trusted. Only the proof speaks the truth.
In (\triangle ABC), (D) is on (BC) such that (AD) is angle bisector. (E) is on (AC) such that (DE \parallel AB). Prove that (AE = EC). 106 geometry problems
Angle bisector theorem: (BD/DC = AB/AC). Take Problem 12
The "Q.E.D." at the bottom of the page— Quod Erat Demonstrandum —is not just a sign-off. It is a sigh of relief. It is the closing of a loop. They don't look like they line up
The second half is where the "AwesomeMath" magic happens. These problems often require multiple "aha!" moments and the use of sophisticated theorems such as: Inversion Homothety Simson Line and Steiner Line properties 3. Why This Book is Different
If stuck for 20 min, switch to coordinates/complex numbers (but only if allowed in contest – IMO accepts pure synthetic or analytic).